Gödel's Incompleteness Theorems
In any system of arithmetic strong enough to be useful, there are true statements it can never prove. Including this one.
193125 min read
Difficulty
Expert
The deep layer is genuinely technical.
Written for: Graduate level or specialist readers.
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The Banach–Tarski Paradox
Cut a solid ball into five pieces, move them around, and you have two balls the same size as the first. Nothing was stretched.
192423 min read
Arrow's Impossibility Theorem
There is no fair way to turn what a group of people each want into what the group wants. This is not a complaint about politics. It is a theorem.
1950 (paper), 1951 (book), 1963 (corrected second edition)25 min read
The Black Hole Information Paradox
Physics insists information can never be destroyed. Black holes appear to destroy it. Both of those cannot be true.
1974/75 (Hawking radiation), 1976 (the paradox stated)24 min read
The EPR Paradox
Measure one particle here and you instantly know something about its partner light-years away. Einstein thought this proved physics was unfinished.
1935 (argument), 1964 (Bell's theorem), 2015 (loophole-free tests)21 min read
The FLP Impossibility Result
A handful of computers cannot be guaranteed to agree on a single value if even one of them might be slow. Not merely hard — impossible.
198523 min read